arX

iv:1

007.

4431

v1 [

mat

h-ph

] 2

6 Ju

l 201

0

Orthogonal polynomials of compact simple Lie groups:

Branching rules for polynomials

M. Nesterenko§, J. Patera†, M. Szajewska‡, and A. Tereszkiewicz‡

§ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., Kyiv-4, 01601, Ukraine;† CRM, Universite de Montreal, C.P.6128-Centre ville, Montreal, Canada;‡ Institute of Mathematics, University of Bialystok, Akademicka 2, PL-15-267, Bialystok, Poland.

E-mail: §maryna@imath.kiev.ua, †patera@crm.umontreal.ca,‡marzena@math.uwb.edu.pl, a.tereszkiewicz@uwb.edu.pl

Polynomials in this paper are defined starting from a compact semisimple Lie group.A known classification of maximal, semisimple subgroups of simple Lie groups is used toselect the cases to be considered here. A general method is presented and all the cases ofrank ≤ 3 are explicitly studied. We derive the polynomials of simple Lie groups B3 andC3 as they are not available elsewhere. The results point to far reaching Lie theoreticalconnections to the theory of multivariable orthogonal polynomials.

1 Introduction

The main purpose of the paper is to demonstrate, describe and illustrate homomorphic relations(also called reduction or branching) between families of polynomials with different underlying Liegroups. The polynomials can be viewed as multivariable generalizations of classical Chebyshevpolynomials of one variable or as subfamilies of multivariable Macdonald polynomials [13]. Thesystematic study of such relations became possible after several families of polynomials in n

variables were constructed [20] via semisimple Lie groups of rank n.The relations studied here are consequences of maximal inclusions of semisimple Lie groups

in simple compact Lie groups. They parallel familiar branching rules for finite dimensionalrepresentations of corresponding Lie algebras (see for example [14]), but cannot be obtainedfrom them in any direct way. The technique exploited in the computation of the branching rulesfor polynomials is the adaptation of the method previously used in [9] and [14]. Related problemsare the computation of branching rules for orbit functions [6, 7] and Weyl group orbits [9].

The problems share certain tools, namely the projection matrices for weights. All the casesof interest to us were classified half a century ago by E.B. Dynkin. In this paper we describethe general principle of the method, and consider all the specific cases with rank n 6 3.

In the literature [20] one finds sufficiently many explicit examples of polynomials for thegroups of rank n = 2, but only those of A3 for n = 3. Therefore we start by computingthe polynomials of the group B3 and C3. As in [20], we take the orbit functions of the threefundamental weights for either of the two Lie groups as our polynomial variables. Such asubstitution imposes transformations of the orthogonality domains. For the orbit functions ofB3 and C3 these were tetrahedra inscribed in a cube. The polynomial substitution transformsthem into the domains F shown in Figures 4 and 5.

New here is description of the continuous and discrete orthogonality of the polynomials withindomains F.

We start from the C-functions of [6] and from the S-functions of [7] (see also [22, 16]), byspecializing them to three variables and converting them into polynomials. The substitution

1

of variables providing the conversion was introduced in [20]. The simplest 1-variable version ofsuch a conversion is found in the context of Chebyshev polynomials [23] and their generalizationto two variables [20], [8]. The background to our work is at the crossroad of the theory ofcompact semisimple Lie groups particularly the properties of their characters, the theory ofspecial functions of mathematical physics, and the orthogonal polynomials of many variables.From the Lie theory we take the uniformity of description of properties of each simple Lie groupand of their characters. The price to pay for that is the need to work with non-orthogonalbases and with character functions of ever increasing complexity the larger the weights get.By working with the orbit functions instead, we circumvent the problem of characters whilestill working with their W -invariant constituents. Contact with the theory of special functionsis made through properties of orbit functions that are symmetric and skew-symmetric withrespect to boundaries of F, called the C- and S-functions respectively [6, 7]. The two familiesof functions have been part of Lie theory for almost a century. Their properties as specialfunctions, particularly their discrete orthogonality, were recognized only a few decades ago [15].Although the characters could also be viewed as special functions of mathematical physics, theircomplexity disqualifies them from almost all applications. To the best of our knowledge theidea to see the root systems as the backbone of the theory of orthogonal polynomials of manyvariables comes from [12]. Here we use the theory of simple Lie groups in order to constructand reduce multivariate orthogonal polynomials. In retrospect, the results of [8] are based ongroup A2, the results of [10] on An. The approach exploits complete reducibility of productsof orbit functions in order to build polynomials from the lowest few. The orbit functions offundamental weights of the Lie group become the polynomial variables. Let us emphasize thatthere are alternatives to our approach that are untested so far. Products of characters are alsocompletely decomposable into their sums. Therefore a similar recursive procedure would buildthe polynomials as functions of variables that are characters of fundamental representations ofthe underlying Lie group. Due to the basic role of the characters, this version of our method canbe preferable for some problems. However, the complex structure of the characters, as opposedto orbit functions, makes it practically more cumbersome. The two approaches coincide forsimple Lie groups An. In such cases the characters of the fundamental weights are equal to theorbit functions of the same weights.

2 Preliminaries

The notions of polynomials under consideration in n variables depend essentially on the under-lying semisimple Lie group G of rank n. This section is intended to fix notation and terminologyand to recall the definitions and some properties of orbit functions. Additional information onthis subject can be found, for example, in [5, 3, 6, 7].

2.1 Bases, Weyl group and orbit functions

Let Rn be the real Euclidean space spanned by the simple roots of a simple Lie group G

(equivalently, Lie algebra). The basis of the simple roots and the basis of fundamental weightsare hereafter referred to as the α-basis and ω-basis, respectively. The two bases are linked bythe Cartan matrix C in the following way α = Cω where

C := (Cjk) =

(2〈αj , αk〉〈αk, αk〉

), hereafter j, k = 1, 2, . . . , n.

The Cartan matrix provides, in principle, all the information needed about G. The same dataabout group G can be taken from the Coxeter-Dynkin diagrams, see e.g. [14].

2

We also use the convention that for the long roots αk the inner product 〈αk, αk〉 = 2, and weintroduce bases dual to α- and ω-bases, denoted here as α- and ω-bases, respectively. The dualbases are fixed by the relations

αj =2αj

〈αj , αj〉, ωj =

2ωj

〈αj , αj〉, 〈αj , ωk〉 = 〈αj , ωk〉 = δjk,

Occasionally it is also useful to work with the orthonormal basis {e1, e2, . . . , en} of Rn.Now we can form the root lattice Q and the weight lattice P of G by all integer linear

combinations of the α-basis and ω-basis,

Q = Zα1 + Zα2 + · · ·+ Zαn, P = Zω1 + Zω2 + · · ·+ Zωn.

In the weight lattice P , we define the cone of dominant weights P+ and its subset of strictlydominant weights P++

P ⊃ P+ = Z≥0ω1 + · · · + Z

≥0ωn ⊃ P++ = Z>0ω1 + · · ·+ Z

>0ωn.

Analogously dual latices Q and P are defined as follows

Q = Zα1 + Zα2 + · · ·+ Zαn, P = Zω1 + Zω2 + · · ·+ Zωn.

Weyl group W (G) is the finite group generated by reflections in (n − 1)-dimensional hyper-planes orthogonal to simple roots, having the origin as their common point, and referred to aselementary reflections rαj

= rj , j = 1, . . . , n.The orbit of W containing the (dominant) point λ ∈ P+ ⊂ R

n is written as Wλ. The sizeof Wλ is denoted by |Wλ| (it is the number of points in Wλ), and order of the Weyl group isdenoted by |W |.

The fundamental region F (G) ⊂ Rn is the convex hull of the vertices {0, ω1

m1, . . . , ωn

mn}, where

mj, j = 1, n are marks of the highest root ξ of the root system.In this paper we mainly deal with the simple Lie groups of rank three, and in the Appendix we

present all necessary information about such groups, i.e., their Cartan matrices, Weyl group or-bits, highest roots, Weyl orbit sizes and fundamental regions. The above brings us to definitionsof symmetric and antisymmetric orbit functions.

Definition 1. The C-function Cλ(x) of G is defined as

Cλ(x) :=∑

µ∈Wλ(G)

e2πi〈µ,x〉, x ∈ Rn, λ ∈ P+.

Definition 2. The S-function Sλ(x) is defined as

Sλ(x) :=∑

µ∈Wλ(G)

(−1)p(µ)e2πi〈µ,x〉, x ∈ Rn, λ ∈ P++ .

where p(µ) is the number of elementary reflections necessary to obtain µ from λ.

The same µ can be obtained by different successions of reflections, but all shortest routesfrom λ to µ will have same parity length, so S-functions are well defined.

In this paper, we always suppose that λ, µ ∈ P are given in ω-basis and x ∈ Rn is given in

α basis, namely λ =n∑

j=1λjωj, µ =

n∑j=1

µjωj, λj, µj ∈ Z and x =n∑

j=1xjαj , xj ∈ R. Therefore

the orbit functions have the following forms

Cλ(x) =∑

µ∈Wλ

e2πi

n∑j=1

µjxj

=∑

µ∈Wλ

n∏

j=1

e2πiµjxj , (1)

3

Sλ(x) =∑

µ∈Wλ

(−1)p(µ)e2πi

n∑j=1

µjxj

=∑

µ∈Wλ

(−1)p(µ)n∏

j=1

e2πiµjxj . (2)

The introduced orbit functions have many useful properties, e.g., continuity, orthogonality,symmetry (antisymmetry) with respect to the boundary of F , eigenfunctions of the differentialoperators, etc. (for details see [6, 7]).

2.2 Discretization of orbit functions

Both C- and S-families of orbit functions are orthogonal and complete, which makes themperfect for the Fourier analysis. As a lattice for the Fourier analysis we choose the refinementof the Z-dual lattice to Q, namely 1

MP , where M ∈ N.

Repeated reflections of F (G) in its (n− 1)-dimensional sides results in tiling the entire spaceRn by copies of F , moreover it is sufficient to consider orbit functions only on the fundamental

region, therefore let us discretize F .We define FM ⊂ F , depending on an arbitrary natural number M as follows

FM =

{s1

Mω1 +

s2

Mω2 + · · ·+ sn

Mωn | s1, . . . , sn ∈ Z

≥0,

n∑

i=1

simi ≤ M ∈ N

}.

For S-functions, the discretized fundamental region is the interior of FM and it has the form

FM =

{s1

Mω1 +

s2

Mω2 + · · ·+ sn

Mωn | s1, . . . , sn ∈ Z

>0,

n∑

i=1

simi ≤ M ∈ N

}.

The number of points of the grid FM (or FM ) is denoted by |FM | (or |FM |).We define the scalar product (see [3]) of two functions f, g : FM (G) → C by

〈f, g〉FM=

∑

x∈FM

ε(x)f(x)g(x), where ε(x) := |Wx|.

The same orthogonality relation holds true for f, g : FM → C.For C- and S-functions normalized by the order of stabilizer of λ, we have:

〈Cλ(x), Cλ′(x)〉FM=

∑

x∈FM

|Wx|Cλ(x)Cλ′(x) = detC|W |2|Wλ|

Mnδλ λ′ , (3)

〈Sλ(x), Sλ′(x)〉FM

=∑

x∈FM

|W |Sλ(x)Sλ′(x) = detC |W | Mnδλ λ′ . (4)

Precise values of |W |, |Wλ|, |FM | and |FM | for the simple Lie groups of rank three arepresented in the Appendix.

3 Orthogonal polynomials in n variables

In this section we fix notations, recall definitions of multivariate polynomials of simple Lie groupsintroduced in [20] and explain some useful notions taken from [2] and [24].

The main objects of this paper are polynomials in n variables

Pk1,...,kn(u)=

k1,...,kn∑

j1,...,jn=0

aj1,...,jnuj11 · · · ujnn , where u := (u1, . . . , un) ∈ C

n, aj1,...,jn ∈ R. (5)

4

Definition 3 (level vector order). Let (a1, a2, . . . , an) be the level vector for G (see [1]) and letn = (a1, a2, . . . , an)(k1, k2, . . . , kn)

t, and n′ = (a1, a2, . . . , an)(k′1, k

′2, . . . , k

′n)

t.Then we say that

uk11 uk22 · · · uknn ≻ uk′11 u

k′22 · · · uk′nn

if n > n′ or if n = n′ and the first nonzero entry in the n-tuple (k1−k′1, k2−k′2, . . . , kn−k′n) isnegative.

In fact the level vector order for vectors (k1, k2, . . . , kn) and (k′1, k′2, . . . , k

′n) coincides with

the graded lexicographical order for (a1k1, a2k2, . . . , ankn) and (a1k′1, a2k

′2, . . . , ank

′n) vectors.

As soon as the above ordering is fixed, the highest modified total degree of the monomialsuk11 · · · uknn (i.e. max{a1k1+ · · ·+ankn}) of the polynomial Pk1,...,kn(u) is called the modified totaldegree of polynomial.

Hereafter, for cases n = 2 and n = 3, we denote (k1, k2) =: (k, l) and (k1, k2, k3) =: (k, l,m).

The level vectors for Lie algebras of ranks two and three are in the Appendix, for all casessee [14].

Example 1. Consider group B3. Its level vector equals (6, 10, 6). Let us order monomials u21,u22 and u33 in the case of B3. To do this we respectively calculate vectors n, n′ and n′′:

n = (6, 10, 6)(2, 0, 0)t , n′ = (6, 10, 6)(0, 2, 0)t , n′′ = (6, 10, 6)(0, 0, 3)t .

Therefore we obtain u22 ≻ u33 ≻ u21.

Similarly for the few first degrees we have:

u22 ≻ u31 ≻ u21u3 ≻ u1u23 ≻ u33 ≻ u1u2 ≻ u2u3 ≻ u21 ≻ u1u3 ≻ u23 ≻ u2 ≻ u1 ≻ u3 ≻ 1.

Definition 4. Let {Pk1,...,kn(u)} ∈ C[u1, . . . , un] be a family of polynomials in n variablessatisfying

∫

Cn

Pk1,...,kn(x)Pk′1,...,k′

n(x)dρ(x) =

∑

u∈VPk1,...,kn(u)Pk′1,...,k

′

n(u)(u) = Λk1,...,knδk1k′1 · · · δknk′n ,

where dρ(x) =∑u∈V

(x)δ(x− u) is the discrete measure, V is a lattice in Cn and Λk1,...,kn is the

normalization constant (see [2], [24] and [4]).The family {Pk(u)} is then called orthogonal polynomials in n variables.

The orthogonal polynomials have a number of useful properties, in particular each polynomialsatisfies recurrence relation, see, e.g. [24] and [4].

3.1 Orthogonal polynomials of simple group G

Here we use the approach to the construction of orthogonal polynomials in n variables that wasproposed in [20]. It is based on the idea of replacement of the lowest C-orbit functions Cωj

,j = 1, 2, . . . , n by new variables Xj . This method brought us to the generalization of classicalChebyshev polynomials to Chebyshev polynomials in n-dimensional Euclidean space. Moreoverit easily gives us rather wide families of Macdonald polynomials. Polynomials generated fromC-functions can be viewed as the generalized Chebyshev polynomials of the first kind in n vari-ables and as the Macdonald symmetric polynomials for the case kα = 0, tα = 1. S-polynomialsplay role of generalized Chebyshev polynomials of the second kind and equivalent to the Mac-donald polynomials with kα = 1, tα = qα. The S-functions divided by the lowest S-function

5

Sρ(x) coincide with the character of the representation and we do use these fractions as theS-polynomials.

Consider C-functions and S-functions defined in the Preliminaries and introduce new coor-dinates

u1 := C(1,0,...,0)(x), u2 := C(0,1,0,...,0)(x), . . . , un := C(0,...,0,1)(x). (6)

These variables coincide with those introduced in [20]: Xj = Cωj= uj, j = 1, 2, . . . , n. It was

shown in [20] (see Proposition 1) that in these coordinates orbit functions gain the polynomialform, and it directly follows from the orthogonality of orbit functions that polynomials Cλ(u)and Sλ(u) are orthogonal. The orthogonality regions for polynomials Cλ(u) and Sλ(u) are theimages F and F of the fundamental regions F and F under the transformation x = (x1, . . . , xn) 7→u = (u1, . . . , un). The discretization developed for the orbit functions (see Section 2.2) can beeffectively applied to these polynomials. Let us fix positive M and let x = ( s1

M, . . . , sn

M) ∈ FM .

Using (3) we have

〈Cλ(x), Cλ′(x)〉FM=∑

x∈FM

|Wx|Cλ(x)Cλ′(x)=∑

u∈FM

|Wu|J(u)Cλ(u)Cλ′(u)=detC |W |2Mn

|Wλ| δλ λ′ ,

where J−1(u) is the discretized Jacobian of the transformation x 7→ u.

Thereby we obtain discrete orthogonality of polynomials Cλ(u) with the weight function |Wu|J(u).Let us calculate J(u) explicitly. Whereas ρ=ω1+ . . .+ωn = (1, . . . , 1)ω and S(u) := S2

ρ(x),

it is easy to check that Jacobian J(u) = 1

(2π)n√

|S(u)|for An, Bn and Cn. Similarly for the S-

polynomials defined as character Sλ(u) :=Sλ+ρ(x)Sρ(x)

from (4), we have the discrete orthogonality

of characters

〈Sλ(x), Sλ′(x)〉FM

=∑

x∈FM

|W |Sλ(x)Sλ′(x)=∑

u∈FM

|W |J(u)S(u)Sλ(u)Sλ′(u)=detC|W |Mnδλλ′ .

Hereof we obtained discrete orthogonality of polynomials with the weight function |W |J(u)S(u)and J(u) = 1

(2π)n√

|S(u)|for An, Bn and Cn.

Explicit values of the weight functions for the groups A2, C2, A3, B3 and C3 are presentedin Appendix.

4 Polynomials of two complex variables

This section recalls recursion relations and lowest orthogonal polynomials generated from groupsA2 and C2. We need them here in order to recognize the result of the reduction to the max-imal subgroup in Section 6. In contrast to the paper [20], we consider explicit examples ofdiscretization of polynomials and transformed fundamental regions F.

In this section, we mean that S-polynomials are the fractionsS(k+1,l+1)(x)

S(1,1)(x), and we denote

them by S(k,l)(u).

4.1 C- and S-polynomials of A2

Let us introduce new coordinates u := (u1, u2) = (C(1,0)(x), C(0,1)(x)). Using the decompositionof products of orbit functions (see [20]) we obtain the set of recurrence relations. In the case ofS-polynomials we adduce only generic recurrence relations, and the lowest S-polynomials canbe obtained from C-polynomials using the Weyl character formula and multiplicities from [1].The lowest C-polynomials are constructed and arranged in Table 1.

6

Generic recurrence relations for S-polynomials of A2 (Chebyshev polynomials of the secondkind in two variables):

S(k+1,l)(u) = u1S(k,l)(u)− S(k,l−1)(u)− S(k−1,l+1)(u), k, l > 1;

S(k,l+1)(u) = u2S(k,l)(u)− S(k−1,l)(u)− S(k+1,l−1)(u), k, l > 1.

Recurrence relations for C-polynomials of A2 (Chebyshev polynomials of the first kind in twovariables):

C(k+1,l)(u) = u1C(k,l)(u)− C(k,l−1)(u)− C(k−1,l+1)(u), k, l > 1;

C(k,l+1)(u) = u2C(k,l)(u)− C(k+1,l−1)(u)− C(k−1,l)(u), k, l > 1;

C(k+1,0)(u) = u1C(k,0)(u)− C(k−1,1)(u), k > 1;

C(0,l+1)(u) = u2C(0,l)(u)− C(1,l−1)(u), l > 1.

♯0 ♯1 ♯2

C(k,l)(u) 1 u1u2 u31 u32C(0,0)(u) 1

C(1,1)(u) −3 1

C(3,0)(u) 3 −3 1

C(0,3)(u) 3 −3 0 1

C(k,l)(u) u2 u21 u1u22

C(0,1)(u) 1

C(2,0)(u) −2 1

C(1,2)(u) −1 −2 1

C(k,l)(u) u1 u22 u21u2C(1,0)(u) 1

C(0,2)(u) −2 1

C(2,1)(u) −1 −2 1

Table 1: Lowest C(k,l)-polynomials of the Lie group A2 split into three classes in correspondencewith the congruence number # of the parameters (k, l).

Let us study the transformation (6) of the fundamental region F (A2) // F(A2). Duringthe substitution x 7→ u, the vertices of the simplex F (A2) (see Appendix) go to the points{P0, P1, P2} and edges go to the continuous curves. Explicitly we have

(0, 0) 7→ (3, 0) =: P0; ω1 7→ (−32 ,−3

√3

2 ) =: P1; ω2 7→ (−32 ,

3√3

2 ) =: P2.

Example 2. Let us fix M = 3. |F3(A2)| = 10 and the corresponding grid points ( s1M, s2M) in

coordinates (Re(u1), Im(u1)) are

(0, 0) 7→ (3, 0); (0, 1) 7→ (−32 ,

3√3

2 );

(23 ,13) 7→ (−2 cos π

9+cos 2π9 ,−2 sin π

9− sin 2π9 ); (1, 0) 7→ (−3

2 ,−3√3

2 );(23 , 0) 7→ (− cos π

9+2 sin π18 ,−2 cos π

18+sin π9 ); (13 ,

23) 7→ (−2 cos π

9+cos 2π9 , 2 sin π

9+sin 2π9 );

(13 , 0) 7→ (2 cos 2π9 +sin π

18 , cosπ18−2 sin 2π

9 ); (0, 13 ) 7→ (2 cos 2π9 +sin π

18 ,− cos π18+2 sin 2π

9 );(13 ,

13) 7→ (0, 0); (0, 23 ) 7→ (−2 cos π

9+2 sin π18 , 2 cos

π18− sin π

9 ).

The choice of new coordinates in the form (Re(u1), Im(u1)) is determined by the complex con-jugation u1 = u2. Plotting these points and the transformed fundamental region F in Figure 1,we see that the discrete set FM (A2) in new coordinates (u1, u2) goes to points FM (A2) ⊂ F(A2).

4.2 C- and S-polynomials of C2

As in the previous section, we obtain by the same substitution the orthogonal polynomials ofgroup C2. Recursion relations for these S- and C-polynomials are also obtained by the expansionof the products of orbit functions.

7

-1 1 2 3

-2

-1

1

2

Figure 1: Region of orthogonality F of polynomials of A2 and discrete points of F3 obtained inExample 2.

The generic recurrence relations for C-polynomials are the following

C(k+1,l)(u) = u1C(k,l)(u)− C(k−1,l)(u)− C(k+1,l−1)(u)− C(k−1,l+1)(u), k, l > 1;

C(k,l+1)(u) = u2C(k,l)(u)− C(k,l−1)(u)− C(k+2,l−1)(u)− C(k−2,l+1)(u), k > 2, l > 1.

Some of the additional relations are:

C(k+1,0)(u) = u1C(k,0)(u)− C(k−1,0)(u)− C(k−1,1)(u), k > 1;

C(0,l+1)(u) = u2C(0,l)(u)− C(0,l−1)(u)− C(2,l−1)(u), l > 1.

The remaining low-order C-polynomials necessary to solve all above recursions are presented inTable 1.

When we have the lowest C-polynomials, all S-polynomials can be found from the relations:

S(k+1,l)(u) = u1S(k,l)(u)− S(k+1,l−1)(u)− S(k−1,l+1(u)− S(k−1,l)(u), k, l > 1;

Sk〈+1(u) = u2S(k,l)(u)− S(k,l−1)(u)− S(k+2,l−1)(u)− S(k−2,l+1)(u), k > 2, l > 1.

C(k,l)(u) 1 u2 u21 u22 u21u2 u32C(0,0)(u) 1

C(0,1)(u) 0 1

C(2,0)(u) −4 −2 1

C(0,2)(u) 4 4 −2 1

C(2,1)(u) 0 −6 0 −2 1

C(0,3)(u) 0 9 0 6 −3 1

C(k,l)(u) u1 u1u2 u31 u1u22

C(1,0)(u) 1

C(1,1)(u) −2 1

C(3,0)(u) −3 −3 1

C(1,2)(u) 6 3 −2 1

Table 2: Lowest C-polynomials of C2 split into two congruence classes according to the parityof the parameter k.

Under the transformation x 7→ u, the vertices of the simplex F (C2) (see Appendix) go topoints {P0, P1, P2}, namely (0, 0) 7→ (4, 4) =: P0, ω1 7→ (0,−4) =: P1, ω2 7→ (−4, 4) =: P2.

8

Example 3. Let us fix M = 4. |F4(C2)| = 9 and the corresponding grid points ( s1M, s2M) in

coordinates (u1, u2) are:

(0, 0) 7→ (4, 4), (0, 14 ) 7→ (2√2, 2), (12 , 0) 7→ (0,−4),

(0, 12) 7→ (0, 0), (0, 34 ) 7→ (−2√2, 2), (14 ,

12 ) 7→ (−2, 0),

(0, 1) 7→ (−4, 4), (14 , 0) 7→ (2, 0), (14 ,14 ) 7→ (0,−2).

In Figure 2 we see the discrete set FM (C2) in new coordinates (u1, u2) and new fundamentalregion F(C2).

-4 -2 2 4

-4

-2

2

4

Figure 2: Region of orthogonality F and discrete points F4 of polynomials of C2.

5 Polynomials in three variables

In this section, we first recall recursion relations and orthogonal polynomials of group A3 andthen we present new orthogonal polynomials of groups B3 and C3 together with the completesets of recurrence relations. The orthogonality domains of the polynomials of all these groupsand discretization examples are shown.

For the orbit functions of each of the groups A3, B3 and C3, we introduce the new coordinatesby the rule x 7→ u, where x = (x1, x2, x3), u = (u1, u2, u3) and

u1 := Cω1(x) = C(1,0,0)(x), u2 := Cω2(x) = C(0,1,0)(x), u3 := Cω3(x) = C(0,0,1)(x). (7)

Note that for different groups of rank three, the lowest C-functions u1, u2, u3 have differentforms and properties, but the general substitution rule (7) is always the same.

The recurrence relations for C- and S- polynomials come from the expansions of the productsujCλ and ujSλ, j = 1, 2, 3.

We also mean that S-polynomials are the fractionsSλ+ρ(x)Sρ(x)

and we denote them as Sλ(u).

As in this section λ has only three coordinates in ω-basis, we use the notations λ = (k, l,m),C(k,l,m)(u) and S(k,l,m)(u).

5.1 C- and S-polynomials of A3 in three complex variables

When the lowest C-functions Cω1(x), Cω2(x) and Cω3(x) of A3 are written explicitly, it is easyto verify that u2 = u2 and u3 = u1.

Generic recursions for C-polynomials (Chebyshev polynomials of the first kind) in threecomplex variables are:

C(k+1,l,m)(u) = u1C(k,l,m)(u)−C(k−1,l+1,m)(u)−C(k,l,m−1)(u)−C(k,l−1,m+1)(u), k, l,m > 1;

9

C(k,l+1,m)(u) = u2C(k,l,m)(u)−C(k+1,l,m−1)(u)−C(k+1,l−1,m+1)(u)

−C(k−1,l+1,m−1)(u)−C(k−1,l,m+1)(u)−C(k,l−1,m)(u), k, l,m > 1;

C(k,l,m+1)(u) = u3C(k,l,m)(u)−C(k+1,l−1,m)(u)−C(k−1,l,m)(u)−C(k,l+1,m−1)(u), k, l,m > 1.

Note that the last of the above relations can also be obtained from C(k,l,m)(u) = C(m,l,k)(u)for k, l,m ∈ Z

>0.

Additional recursion relations for C-polynomials of A3:

C(k+1,0,0)(u) = u1C(k,0,0)(u)− C(k−1,1,0)(u), k > 1;

C(0,k+1,0)(u) = u2C(0,k,0)(u)− C(0,k−1,0)(u)− C(1,k−1,1)(u), k > 1;

C(k+1,l,0)(u) = u1C(k,l,0)(u)−C(k,l−1,1)(u)− C(k−1,l+1,0)(u), k, l > 1;

C(k,l+1,0)(u) = u2C(k,l,0)(u)−C(k−1,l,1)(u)− C(k,l−1,0)(u)− C(k+1,l−1,1)(u), k, l > 1;

C(k+1,0,m)(u) = u1C(k,0,m)(u)− C(k−1,1,m)(u)− C(k,0,m−1)(u) k,m > 1.

The remaining low-order C-polynomials of A3 can be found in Table 3.

♯0 ♯1

C(k,l,m)(u) 1 u1u3 u22 u2u23 u21u2

C(0,0,0)(u) 1

C(1,0,1)(u) −4 1

C(0,2,0)(u) 2 −2 1

C(0,1,2)(u) 4 −1 −2 1

C(2,1,0)(u) 4 −1 −2 0 1

C(k,l,m)(u) u1 u2u3 u33 u21u3 u1u22

C(1,0,0)(u) 1

C(0,1,1)(u) −3 1

C(0,0,3)(u) 3 −3 1

C(2,0,1)(u) −1 −2 0 1

C(1,2,0)(u) 5 −1 0 −2 1

♯2 ♯3

C(k,l,m)(u) u2 u23 u21 u1u2u3 u32C(0,1,0)(u) 1

C(0,0,2)(u) −2 1

C(2,0,0)(u) −2 0 1

C(1,1,1)(u) 4 −3 −3 1

C(0,3,0)(u) −3 3 3 −3 1

C(k,l,m)(u) u3 u1u2 u31 u1u23 u22u3

C(0,0,1)(u) 1

C(1,1,0)(u) −3 1

C(3,0,0)(u) 3 −3 1

C(1,0,2)(u) −1 −2 0 1

C(0,2,1)(u) 5 −1 0 −2 1

Table 3: Lowest C-polynomials of A3 split into four congruence classes # = 0, # = 1, # = 2and # = 3.

Let us study the transformation of the fundamental region F (A3) //F(A3). During the sub-stitution x 7→ u the vertices of the simplex F (A3) (see Appendix) go to the points {P0, P1, P2, P3}:

(0, 0, 0) 7→ (4, 6, 0) =: P0, ω1 7→ (0,−6,−4) =: P1,

ω2 7→ (−4, 6, 0) =: P2, ω3 7→ (0,−6, 4) =: P3.

The shape of the region of orthogonality of polynomials of A3 is presented in Figure 3.

Example 4. Let us consider discretization with M = 3. |F3(A3)| = 20 and grid points( s1M, s2M, s3M) in new coordinates (Re(u1), u2, Im(u1)) are:

10

(0, 0, 0) 7→ (4, 6, 0), (0, 0, 1) 7→ (0,−6, 4), (0, 1, 0) 7→ (−4, 6, 0),

(23 , 0,13) 7→ (0,−3,−2), (23 ,

13 , 0) 7→ (−1

2 ,−3,−3√3

2 ), (13 , 0,23) 7→ (0,−3, 2),

(13 ,23 , 0) 7→ (−3

√3

2 , 3,−12 ), (0, 23 ,

13 ) 7→ (−3

√3

2 , 3, 12), (13 , 0, 0) 7→ (3√3

2 , 3,−12 ),

(0, 13 , 0) 7→ (2, 3, 0), (13 ,13 , 0) 7→ (0, 0,−1), (13 , 0,

13) 7→ (1, 0, 0),

(13 ,13 ,

13 ) 7→ (−1, 0, 0), (23 , 0, 0) 7→ (12 ,−3,−3

√3

2 ), (0, 0, 23) 7→ (12 ,−3, 3√3

2 ),

(1, 0, 0) 7→ (0,−6,−4), (0, 13 ,23 ) 7→ (−1

2 ,−3, 3√3

2 ), (0, 0, 13) 7→ (3√3

2 , 3, 12),(0, 13 ,

13) 7→ (0, 0, 1), (0, 23 , 0) 7→ (−2, 3, 0).

All these points belong to the new fundamental region F(A3).

-4

-2

0

2

4

-5

0

5

-4

-2

0

2

4

Figure 3: Region of orthogonality F and discrete points F3 of polynomials of A3.

Using the Weyl character formula and multiplicities from [1], we can obtain S-polynomials(Chebyshev polynomials of the second kind) in u1, u2 and u3. Or, alternately, having the lowestS-polynomials we can construct other polynomials by means of the following recursions

S(k+1,l,m)(u) = u1S(k,l,m)(u)−S(k−1,l+1,m)(u)−S(k,l,m−1)(u)−S(k,l−1,m+1)(u), k, l,m > 2;

S(k,l+1,m)(u) = u2S(k,l,m)(u)−S(k,l−1,m)(u)−S(k+1,l−1,m+1)(u)−S(k−1,l+1,m−1)(u)

−S(k+1,l,m−1)(u)−S(k−1,l,m+1)(u), k, l,m > 2;

S(k,l,m+1)(u) = u3S(k,l,m)(u)−S(k+1,l−1,m)(u)−S(k−1,l,m)(u)−S(k,l+1,m−1)(u), k, l,m > 2.

5.2 C- and S-polynomials of B3 in three real variables

For group B3, our new coordinates u satisfy the relation ui = ui, i = 1, 2, 3.The following generic recursion relations for C-polynomials hold true when k, l,m > 2:

C(k+1,l,m)(u) = u1C(k,l,m)(u)−C(k,l+1,m−2)(u)−C(k,l−1,m+2)(u)−C(k+1,l−1,m)(u)

−C(k−1,l+1,m)(u)−C(k−1,l,m)(u)−C(k−1,l+1,m)(u)−C(k−1,l,m)(u),

C(k,l+1,m)(u) = u2C(k,l,m)(u)−C(k+1,l−1,m+2)(u)−C(k−1,l−1,m+2)(u)−C(k+1,l−2,m+2)(u)

11

−C(k−1,l,m+2)(u)−C(k−1,l+2,m−2)(u)−C(k+1,l+1,m−2)(u)

−C(k−1,l+1,m−2)(u)−C(k+1,l,m−2)(u)−C(k−2,l+1,m)(u)−C(k+2,l−1,m)(u),

C(k,l,m+1)(u) = u3C(k,l,m)(u)−C(k+1,l−1,m+1)(u)−C(k,l−1,m+1)(u)−C(k−1,l,m+1)(u)

−C(k−1,l+1,m−1)(u)−C(k,l+1,m−1)(u)−C(k+1,l,m−1)(u)−C(k,l,m−1)(u)

Remaining recurrence relations except for the lowest polynomials are listed below:

C(k+1,l,0)(u) = u1C(k,l,0)(u)−C(k−1,l+1,)0(u)−C(k+1,l−1,0)(u)−C(k−1,l,0)(u)−C(k,l−1,2)(u),

C(k,l+1,0)(u) = u2C(k,l,0)(u)−C(k−2,l+1,0)(u)−C(k+2,l−1,0)(u)−C(k−1,l,2)(u)−C(k,l−1,0)(u)

−C(k+1,l−1,2)(u)−C(k−1,l−1,2)(u)−C(k+1,l−2,2)(u),

C(k+1,0,0)(u) = u1C(k,0,0)(u)−C(k−1,1,0)(u)−C(k−1,0,0)(u),

C(0,l+1,0)(u) = u2C(0,l,0)(u)−C(2,l−1,0)(u)−C(0,l−1,0)(u)−C(1,l−1,2)(u)−C(1,l−2,2)(u),

C(0,0,m+1)(u) = u3C(0,0,m)(u)−C(0,1,m−1)(u)−C(1,0,m−1)(u)−C(0,0,m−1)(u),

C(k+1,0,m)(u) = u1C(k,0,m)(u)−C(k,1,m−2)(u)−C(k−1,1,m)(u),

C(k,0,m+1)(u) = u3C(k,0,m)(u)−C(k−1,0,m+1)(u)−C(k−1,1,m−1)(u)−C(k,1,m−1)(u)

−C(k,0,m−1)(u)−C(k+1,0,m−1)(u),

C(0,l+1,m)(u) = u2C(0,l,m)(u)−C(1,l−1,m+2)(u)−C(1,l−2,m+2)(u)−C(0,l−1,m)(u)

−C(2,l−1,m)(u)−C(1,l+1,m−2)(u)−C(1,l,m−2)(u),

C(0,l,m+1)(u) = u3C(0,l,m)(u)−C(1,l−1,m+1)(u)−C(0,l−1,m+1)(u)−C(0,l+1,m−1)(u)

−C(0,l,m−1)(u)−C(1,l,m−1)(u).

The lowest C-polynomials of B3 were calculated explicitly and arranged in Table 4.

As in the previous case, we can use the Weyl character formula or generic recurrence relationsfor S-polynomials of B3 valid for k, l,m > 2:

S(k+1,l,m)(u) = u1S(k,l,m)(u)−S(k,l+1,m−2)(u)−S(k,l−1,m+2)(u)−S(k+1,l−1,m)(u)

−S(k−1,l+1,m)(u)−S(k−1,l,m)(u)−S(k−1,l+1,m)(u)−S(k−1,l,m)(u),

S(k,l+1,m)(u) = u2S(k,l,m)(u)−S(k+1,l−1,m+2)(u)−S(k−1,l−1,m+2)(u)−S(k+1,l−2,m+2)(u)

−S(k+1,l,m−2)(u)−S(k−2,l+1,m)(u)−S(k−1,l,m+2)(u)−S(k−1,l+2,m−2)(u)

−S(k+1,l+1,m−2)(u)−S(k−1,l+1,m−2)(u)−S(k+2,l−1,m)(u),

S(k,l,m+1)(u) = u3S(k,l,m)(u)−S(k+1,l−1,m+1)(u)−S(k,l−1,m+1)(u)−S(k−1,l,m+1)(u)

−S(k−1,l+1,m−1)(u)−S(k,l+1,m−1)(u)−S(k+1,l,m−1)(u)−S(k,l,m−1)(u).

Substitution of variables x 7→ u transforms vertices of the simplex F (B3) into vertices of theorthogonality domain F(B3) = {P0, P1, P2, P3} as follows:

(0, 0, 0) 7→ (6, 12, 8) =: P0, ω1 7→ (6, 12,−8) =: P1,12ω2 7→ (−2,−4, 0) =: P2, ω3 7→ (−6, 12, 0) =: P3.

The domain F(B3) and discretization points from Example 5 are plotted in Figure 4.

12

♯0

C(k,l,m)(u) 1 u1 u2 u21 u23 u1u2 u1u23 u31 u22 u2u

23 u21u2

C(0,0,0)(u) 1

C(1,0,0)(u) 0 1

C(0,1,0)(u) 0 0 1

C(2,0,0)(u) −6 0 −2 1

C(0,0,2)(u) −8 −4 −2 0 1

C(1,1,0)(u) 24 8 6 0 −3 1

C(1,0,2)(u) 0 −8 −2 −4 0 −2 1

C(3,0,0)(u) −24 −15 −6 0 3 −3 0 1

C(0,2,0)(u) 12 16 8 4 0 4 −2 0 1

C(0,1,2)(u) −48 −20 −20 0 6 −6 0 0 −2 1

C(2,1,0)(u) 0 8 −6 4 0 2 −1 0 −2 0 1

♯1

C(k,l,m)(u) u3 u1u3 u2u3 u33 u21u3 u1u2u3C(0,0,1)(u) 1

C(1,0,1)(u) −3 1

C(0,1,1)(u) 3 −2 1

C(0,0,3)(u) −9 −3 −3 1

C(2,0,1)(u) −3 −1 −2 0 1

C(1,1,1)(u) 30 12 8 −3 −2 1

Table 4: Lowest C-polynomials of B3 split into two congruence classes # = 0 and # = 1.

Example 5. Let us fix M = 4. |F4(B3)| = 14 and lattice points ( s1M, s2M, s3M) in new coordinates

(u1, u2, u3) have the form:

(0, 0, 0) 7→ (6, 12, 8), (0, 0, 14 ) 7→ (0, 0, 2√2), (0, 12 , 0) 7→ (−2,−4, 0),

(12 ,14 , 0) 7→ (2, 0,−4), (1, 0, 0) 7→ (6, 12,−8), (34 , 0, 0) 7→ (4, 4,−4

√2),

(0, 14 , 0) 7→ (2, 0, 4), (14 , 0, 0) 7→ (4, 4, 4√2), (14 ,

14 , 0) 7→ (0,−4, 0),

(12 , 0, 0) 7→ (2,−4, 0), (12 , 0,14) 7→ (0, 0,−2

√2), (14 , 0,

14) 7→ (−2, 0, 0),

(0, 14 ,14) 7→ (−4, 4, 0), (0, 0, 12 ) 7→ (−6, 12, 0).

5.3 C- and S-polynomials of C3 in three real variables

Low order C-polynomials of group C3 are listed in Table 5. Higher-order polynomials can beobtained from the recurrence relations. Generic recursions for k, l,m > 2 are:

C(k+1,l,m)(u) = u1C(k,l,m)(u)−C(k,l−1,m+1)(u)−C(k+1,l−1,m)(u)−C(k−1,l+1,m)(u)

−C(k−1,l,m)(u)−C(k,l+1,m−1)(u),

C(k,l+1,m)(u) = u2C(k,l,m)(u)−C(k+1,l,m−1)(u)−C(k−1,l,m+1)(u)−C(k,l−1,m)(u)

−C(k+1,l+1,m−1)(u)−C(k−2,l+1,m)(u)−C(k−1,l−1,m+1)(u)−C(k−1,l+1,m−1)(u)

−C(k+2,l−1,m)(u)−C(k+1,l−1,m+1)(u)−C(k+1,l−2,m+1)(u)−C(k−1,l+2,m−1)(u),

C(k,l,m+1)(u) = u3C(k,l,m)(u)−C(k,l−2,m+1)(u)−C(k−2,l,m+1)(u)−C(k,l+2,m−1)(u)

−C(k,l,m−1)(u)−C(k+2,l−2,m+1)(u)−C(k−2,l+2,m−1)(u)−C(k+2,l,m−1)(u).

13

-5

0

5

0

5

10

-5

0

5

Figure 4: Region of orthogonality F and discrete points F4 of polynomials of B3.

Additional recursions:

C(k+1,0,0)(u) = u1C(k,0,0)(u)−C(k−1,1,0)(u)−C(k−1,0,0)(u), k > 1;

C(k+1,l,0)(u) = u1C(k,l,0)(u)−C(k,l−1,1)(u)−C(k−1,l.0)(u)

−C(k+1,l−1,0)(u)−C(k−1,l+1,0)(u), k, l > 1;

C(k,l+1,0)(u) = u2C(k,l,0)(u)−C(k−1,l,1)(u)−C(k,l−1,0)(u)−C(k+2,l−1,0)(u)−C(k−2,l+1,0)(u)

−C(k−1,l−1,1)(u)−C(k+1,l−1,1)(u)−C(k+1,l−2,1)(u), k, l > 2;

C(0,l+1,m)(u) = u2C(0,l,m)(u)−C(1,l,m−1)(u)−C(0,l−1,m)(u)−C(2,l−1,m)(u)−C(1,l+1,m−1)(u)

−C(1,l−1,m+1)(u)−C(1,l−2,m+1)(u), l > 2, m > 1;

C(0,l,m+1)(u) = u3C(0,l,m)(u)−C(0,l−2,m+1)(u)−C(0,l,m−1)(u)−C(2,l−2,m+1)(u)

−C(2,l,m−1)(u)−C(0,l+2,m−1)(u), l > 2, m > 1;

C(k+1,0,m)(u) = u1C(k,0,m)(u)−C(k−1,1,m)(u)−C(k,1,m−1)(u)−C(k−1,0,m)(u), k,m > 1;

C(k,0,m+1)(u) = u3C(k,0,m)(u)−C(k−2,0,m+1)(u)−C(k,0,m−1)(u)−C(k−2,2,m−1)(u)

−C(k+2,0,m−1)(u)−C(k,2,m−1)(u), k > 2, m > 1;

C(0,l+1,0)(u) = u2C(0,l,0)(u)−C(0,l−1,0)(u)−C(2,l−1,0)(u)−C(1,l−1,1)(u)−C(1,l−2,1)(u), l > 2;

C(0,0,m+1)(u) = u3C(0,0,m)(u)−C(0,0,m−1)(u)−C(2,0,m−1)(u)−C(0,2,m−1)(u), m > 1.

Generic recursions for S-polynomials hold true when k, l,m > 2

S(k+1,l,m)(u) = u1S(k,l,m)(u)−S(k,l−1,m+1)(u)−S(k+1,l−1,m)(u)−S(k−1,l+1,m)(u)

−S(k−1,l,m)(u)−S(k,l+1,m−1)(u),

S(k,l+1,m)(u) = u2S(k,l,m)(u)−S(k+1,l,m−1)(u)−S(k−1,l,m+1)(u)−S(k+2,l−1,m)(u)

−S(k−2,l+1,m)(u)−S(k−1,l−1,m+1)(u)−S(k−1,l+1,m−1)(u)−S(k+1,l+1,m−1)(u)

−S(k+1,l−1,m+1)(u)−S(k+1,l−2,m+1)(u)−S(k−1,l+2,m−1)(u)−S(k,l−1,m)(u),

S(k,l,m+1)(u) = u3S(k,l,m)(u)−S(k,l−2,m+1)(u)−S(k−2,l,m+1)(u)−S(k,l+2,m−1)(u)

14

−S(k+2,l−2,m+1)(u)−S(k−2,l+2,m−1)(u)−S(k+2,l,m−1)(u)−S(k,l,m−1)(u).

♯0

C(k,l,m)(u) 1 u2 u21 u1u3 u22 u21u2 u23 u1u2u3 u32 u2u23

C(0,0,0)(u) 1

C(0,1,0)(u) 0 1

C(2,0,0)(u) −6 −2 1

C(1,0,1)(u) 0 −2 0 1

C(0,2,0)(u) 12 8 −4 −2 1

C(2,1,0)(u) 0 −6 0 −1 −2 1

C(0,0,2)(u) −8 −8 4 4 −2 0 1

C(1,1,1)(u) 0 12 0 −4 4 −2 −3 1

C(0,3,0)(u) 0 9 0 3 6 −3 3 −3 1

C(0,1,2)(u) 0 −18 0 3 −12 6 3 3 −2 1

♯1

C(k,l,m)(u) u1 u3 u1u2 u31 u2u3 u21u3 u1u22 u1u

23 u22u3 v33

C(1,0,0)(u) 1

C(0,0,1)(u) 0 1

C(1,1,0)(u) −4 −3 1

C(3,0,0)(u) −3 3 −3 1

C(0,1,1)(u) 4 6 −2 0 1

C(2,0,1)(u) 0 −9 0 0 −2 1

C(1,2,0)(u) 12 −3 9 −4 −1 −2 1

C(1,0,2)(u) −12 −6 −6 4 −1 4 −2 1

C(0,2,1)(u) 0 27 0 0 12 −6 0 −2 1

C(0,0,3)(u) 0 −27 0 0 −18 9 0 6 −3 1

Table 5: Lowest C-polynomials of C3 split into two congruence classes # = 0 and # = 1.

Let us study the transformation of the fundamental region F (C3) //F(C3). During the substi-tution x 7→ u, the vertices of the simplex F (C3) (see Appendix) go to the points {P0, P1, P2, P3}:

(0, 0, 0) 7→ (6, 12, 8) =: P0, ω1 7→ (2,−4,−8) =: P1,

ω2 7→ (−2,−4, 8) =: P2, ω3 7→ (−6, 12,−8) =: P3.

The region of orthogonality of polynomials of C3 is presented in Figure 5.

Example 6. Let us fix M = 4. |F4(C3)| = 14 and lattice points ( s1M, s2M, s3M) ∈ F4 map to

(u1, u2, u3):

(0, 0, 0) 7→ (6, 12, 8), (0, 0, 14) 7→ (3√2, 6, 2

√2), (0, 14 , 0) 7→ (2, 0, 0),

(0, 14 ,14) 7→ (−

√2,−2, 2

√2), (14 , 0,

14) 7→ (

√2,−2,−2

√2), (14 , 0,

12) 7→ (−2, 0, 0),

(14 ,14 , 0) 7→ (0,−4, 0), (0, 12 , 0) 7→ (−2,−4, 8), (0, 14 ,

12) 7→ (−4, 4, 0),

(12 , 0, 0) 7→ (2,−4,−8), (0, 0, 1) 7→ (−6, 12,−8), (0, 0, 12 ) 7→ (0, 0, 0),

(14 , 0, 0) 7→ (4, 4, 0), (0, 0, 34) 7→ (−3√2, 6,−2

√2).

These points are shown on the orthogonality domain F(C3) in Figure 5.

15

-5

0

5

0

5

10

-5

0

5

Figure 5: Region of orthogonality F and discrete points F4 of polynomials of C3.

6 Branching rules for polynomials of A2, C2, G2, A3, B3, and C3.

Consider simple Lie groups of rank ≤ 3. An uncommon transformation (‘branching’) of polyno-mials of groups into the sum of polynomials of their subgroups. We are interested in cases whena complete reduction of all polynomials of the group is achieved, namely when the sum consistsof complete polynomials of the subgroup.

Probably the most interesting class of cases can be identified by the maximal semisimplesubgroups contained in the simple Lie groups. There are the following cases to consider:

A2 ⊃ A1;

C2 ⊃ A1 ×A1, C2 ⊃ A1;

G2 ⊃ A1 ×A1, G2 ⊃ A2, G2 ⊃ A1.

A3 ⊃ C2, A3 ⊃ A1 ×A1;

B3 ⊃ A3, B3 ⊃ A1 ×A1 ×A1, B3 ⊃ G2;

C3 ⊃ C2 ×A1, C3 ⊃ A2, C3 ⊃ A1.

In addition, complete reduction of polynomials takes place when the subgroup is non-semisimplereductive. Three such subgroups are maximal with rank≤ 3, namely A2 ⊃ A1×U1, A3 ⊃ A2×U1,and B3 ⊃ B2 × U1. Here, irreducible polynomials of the 1-parametric group U1 are trivially allmonomials, say Y k, with integer k positive or negative.

Finally, complete reduction of polynomials happens when the relation between the pair ofgroups is not inclusion but subjoining [21]. A classification of all such relations is in [17].

6.1 Polynomials of the Lie algebra A1

It was shown in [19, 20] that there is a one-to-one correspondence between C- and S-polynomialsof A1 and Chebyshev polynomials of the first and second kind respectively. Here we just write

16

down the explicit form of the C-polynomials of A1 (the Chebyshev polynomials of the first kind)traditionally denoted by Tm, m = 0, 1, 2, . . . .

T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1,

Tm+1(x) = 2xTm − Tm−1.

The equivalent set of polynomials can be obtained via the substitution x = y2 , and up to the

scaling factor they are:

T0(y) = 1, T1(y) = y, T2(y) = y2 − 1, T3(y) = y3 − 3y, T4(y) = y4 − 4y2 + 2, . . .

6.2 Reductions of polynomials of rank two simple Lie groups.

In this section, we present explicit forms of projection matrices Pr providing reductions fromthe rank two simple Lie groups G to their maximal subgroups H. For the each set {G ⊃ H,Pr}we obtain the corresponding substitution of variables Xi 7→ fi(Y1, Y2) or Xi 7→ fi(Y ), i = 1, 2that reduces the orthogonal polynomials of G to the polynomials of H. Explicit examples ofC-polynomial reductions are shown for all pairs G ⊃ H.

Here we use the notation of polynomial variables taken from the paper [20], namelyXi := Cωi,

instead of ui = Xi, i = 1, 2 as in previous sections.

Reduction from A2 to A1.

Pr =(2 2

);

C(1,0)(A2) 7→ C(2)(A1),

C(0,1)(A2) 7→ C(2)(A1);therefore

X1 7→ Y 2 − 2,X2 7→ Y 2 − 2.

Example of reduction: Consider the reduction of a few low order polynomials of A2. Hereafterthe explicit forms of polynomials of simple Lie groups of rank two are taken from the paper [20].

C(1,1)(A2) = X1X2 − 3 7→ Y 4 − 4Y 2 + 1 = T4(Y )− Y ;

C(0,2)(A2) = X22 − 2X1 7→ Y 4 − 6Y 2 + 8 = T4(Y )− 2T2(Y ) + 4Y.

Reduction from C2 to A1 ×A1 =: A(1)1 ×A

(2)1 .

Pr =

(1 10 1

);

C(1,0)(C2) 7→ C(1)(A(1)1 )C(0)(A

(2)1 ),

C(0,1)(C2) 7→ C(0)(A(1)1 )C(1)(A

(2)1 );

thereforeX1 7→ Y1,

X2 7→ Y2.

Example of reduction: In this example, Chebyshev polynomials Tm(Y1), m = 1, 2, . . . corre-

spond to the Lie group A(1)1 , and Tm(Y2) correspond to group A

(2)1 .

C(2,1)(C2)=X21X2−2X2

2−6X2 7→ Y 21 Y2−2Y 2

2 −6Y2 =(T2(Y1)−5

)Y2−2

(T2(Y2)+Y2

);

C(3,0)(C2) = X31 − 3X1X2 − 3X1 7→ Y 3

1 − 3Y1Y2 − 3Y1 = T3(Y1)− 3Y1Y2.

17

Reduction from C2 to A1.

Pr =(3 4

);

C(1,0)(C2) 7→ C(3)(A1),

C(0,1)(C2) 7→ C(4)(A1);therefore

X1 7→ Y 3 − 3Y,X2 7→ Y 4 − 4Y 2 + 2.

Example of reduction:

C(1,1)(C2) = X1X2 − 2X1 7→ Y 7 − 7Y 5 + 12Y 3 = T7(Y )− 2T3(Y ) + Y ;

C(0,2)(C2) = X21 − 2X1 − 4 7→ Y 6 − 8Y 4 + 17Y 2 − 8 = T6(Y )− 2T4(Y )− 2.

Reduction from G2 to A2.

Pr =

(1 01 1

);

C(1,0)(G2) 7→ C(1,1)(A2),

C(0,1)(G2) 7→ C(0,1)(A2);therefore

X1 7→ Y1Y2 − 3,X2 7→ Y2.

Example of reduction: Below C(a,b)(Y ) := C(a,b)(Y1, Y2) are the polynomials of A2.

C(0,2)(G2) = X22−2X2−2X1−6 7→ Y 2

2 −2Y1Y2−2Y2 = C(0,2)(Y )−2C(1,1)(Y )−2Y2+2Y1−6;

C(1,1)(A2) = −2X22+X1X2+2X2+4X1+12 7→

−2Y 22 +Y1Y

22 −Y2+4Y1Y2 = C(1,2)(Y )+4C(1,1)(Y )−2C(0,2)(Y )+2C(2,0)(Y )−4Y1+4Y2+12.

Reduction from G2 to A1 ×A1 =: A(1)1 ×A

(2)1 .

Pr =

(1 10 1

);

C(1,0)(G2) 7→ C(1)(A(1)1 )C(3)(A

(2)1 ),

C(0,1)(G2) 7→ C(0)(A(1)1 )C(2)(A

(2)1 );

thereforeX1 7→ Y1(Y

32 − 3Y2),

X2 7→ Y 22 − 2.

Example of reduction: Here each polynomial Tm(Y1), m = 1, 2, . . . corresponds to A(1)1 and

Tm(Y2) does to A(2)1 .

C(0,2)(G2) = X22−2X2−2X1−6 7→

Y 42 −2Y1Y

32 −6Y 2

2 +3Y1Y2+2 = T4(Y2)−2T2(Y2)−Y1

(2T3(Y2)−3T2(Y2)

)− 4;

C(1,1)(A2) = −2X22+X1X2+2X2+4X1+12 7→

Y1

(Y 52 − Y 3

2 − 6Y2

)− 2Y 4

2 + 10Y 22 = Y1

(T5(Y2)+4T3(Y2) + Y2

)−2

(T4(Y2)−T2(Y2)

)+8.

Reduction from G2 to A1.

Pr =(6 4

);

C(1,0)(G2) 7→ C(6)(A1),

C(0,1)(G2) 7→ C(4)(A1);therefore

X1 7→ Y 6 − 6Y 4 + 9Y 2 − 2,X2 7→ Y 4 − 4Y 2 + 2.

18

Example of reduction: Because the polynomials of G2 and the variable transformations are toocumbersome in this case we present only one example of polynomial reduction.

C(0,2)(G2) = X22−2X2−2X1−6 7→ Y 8−10Y 6+28Y 4−18Y 2−6=T8(Y )−2T6(Y )−4T4(Y )−4.

6.3 Reductions of polynomials of rank three simple Lie groups.

In this section, we obtain the projection matrices Pr providing reductions from the rank threesimple Lie groups G to their maximal subgroupsH. For each set {G ⊃ H,Pr} we also obtain thecorresponding substitution of variables Xi 7→ fi(Y1, Y2, Y3) or Xi 7→ fi(Y1, Y2), or Xi 7→ fi(Y ),i = 1, 2, 3. This reduces the orthogonal polynomials of G to the polynomials of H. Explicitexamples of C-polynomial reductions are shown for all pairs G ⊃ H.

Note that in this section we use the notation of polynomial variables taken from the paper [20],namely Xi := Cωi

, instead of ui = Xi, i = 1, 2, 3 as in the beginning of the paper.

Reduction from A3 to C2.

Pr =

(1 0 10 1 0

);

C(1,0,0)(A3) 7→ C(1,0)(C2),

C(0,1,0)(A3) 7→ C(0,1)(C2),

C(0,0,1)(A3) 7→ C(1,0)(C2);therefore

X1 7→ Y1,

X2 7→ Y2,

X3 7→ Y1.

Example of reduction: Below C(a,b)(Y ) := C(a,b)(Y1, Y2) are the polynomials of the subalge-bra C2.

C(1,0,1)(A3) = −4 +X1X3 7→ −4 + Y 21 = C(2,0)(Y ) + 2Y2;

C(0,1,1)(A3) = −3X1 +X2X3 7→ −3Y1 + Y1Y2 = C(1,1)(Y )− Y1.

Reduction from A3 to A1 ×A1 =: A(1)1 ×A

(2)1 .

Pr =

(1 0 11 2 1

);

C(1,0,0)(A3) 7→ C1(A(1)1 )C1(A

(2)1 ),

C(0,1,0)(A3) 7→ C0(A(1)1 )C2(A

(2)1 ),

C(0,0,1)(A3) 7→ C1(A(1)1 )C1(A

(2)1 );

thereforeX1 7→ Y1Y2,

X2 7→ Y 22 − 2,

X3 7→ Y1Y2.

Example of reduction: In this example, Chebyshev polynomials Tm(Yi), m = 1, 2, . . . corre-

spond to the Lie group A(i)1 , i = 1, 2.

C(1,0,1) = −4 +X1X3 7→ −4 + Y 21 Y

22 = T2(Y1)T2(Y2) + 2T2(Y1) + 2T2(Y2);

C(0,1,1) = −3X1 +X2X3 7→ −5Y1Y2 + Y1Y32 = Y1T3(Y2)− 2Y1Y2.

19

Reduction from B3 to A3.

Pr =

0 1 11 1 00 1 0

;

C(1,0,0)(B3) 7→ C(0,1,0)(A3),

C(0,1,0)(B3) 7→ C(1,1,1)(A3),

C(0,0,1)(B3) 7→ C(1,0,0)(A3);therefore

X1 7→ Y2,

X2 7→ Y1Y2Y3 − 3Y 21 − 3Y 2

3 + 4Y2,

X3 7→ Y1.

Example of reduction: Below C(a,b,c)(Y ) := C(a,b,c)(Y1, Y2, Y3) are polynomials of the subalgebraA3.

C(1,0,1)(B3) = X1X3 − 3X3 7→ Y1Y2 − 3Y1 = C(1,1,0)(Y )− 3Y1 + 3Y3;

C(0,0,2)(B3) = X23 − 2X2 − 4X1 − 8 7→

6Y 23 − 2Y1Y2Y3 − 12Y2 + 7Y 2

1 − 8 = −2C(1,1,1)(Y ) + C(2,0,0)(Y )− 2Y2 − 8.

Reduction from B3 to 3A1 =: A(1)1 ×A

(2)1 ×A

(3)1 .

Pr =

1 1 01 1 10 2 1

;

C(1,0,0)(B3) 7→ C1(A11)C1(A

21)C0(A

31),

C(0,1,0)(B3) 7→ C1(A11)C1(A

21)C2(A

31),

C(0,0,1)(B3) 7→ C0(A11)C1(A

21)C1(A

31);

thereforeX1 7→ Y1Y2,

X2 7→ Y1Y2(Y23 − 2),

X3 7→ Y2Y3.

Example of reduction: The variables refer to the three different subalgebras A1. The Chebyshev

polynomials Tm(Yi), m = 1, 2, . . . correspond to the Lie group A(i)1 , i = 1, 2, 3.

C(1,0,1)(B3) = X1X3 − 3X3 7→ Y1Y22 Y3 − 3Y2Y3 = Y1T2(Y2)Y3 + 2Y1Y3 − 3Y2Y3;

C(0,1,1)(B3) = X2X3 − 2X1X3 + 3X3 7→Y1Y

22 Y

33 − 4Y1Y

22 Y3 + 3Y2Y3 = Y1T2(Y2)T3(Y3) + 3Y1T2(Y2)Y3 + 2Y1T3(Y3) + 6Y1Y3 + 3Y2Y3.

Reduction from B3 to G2.

Pr =

(0 1 01 0 1

);

C(1,0,0)(B3) 7→ C(0,1)(G2),

C(0,1,0)(B3) 7→ C(1,0)(G2),

C(0,0,1)(B3) 7→ C(0,1)(G2);

thereforeX1 7→ Y2,

X2 7→ Y1,

X3 7→ Y2.

Example of reduction: In this example, C(a,b,c)(Y ) := C(a,b)(Y1, Y2) are the polynomials ofsubalgebra G2.

C(1,0,1)(B3) = X1X3 − 3X3 7→ Y 22 − 3Y2 = C(0,2)(Y )− Y2 + 2Y1 + 6;

C(1,1,0)(B3) = X21 − 2X2 − 6 7→ Y 2

2 − 2Y1 − 4Y2 − 8 = C(0,2)(Y ) + Y2.

20

Reduction from C3 to C2 ×A1.

Pr =

1 0 00 1 10 0 1

;

C(1,0,0)(C3) 7→ C(1,0)(G2)C0(A1),

C(0,1,0)(C3) 7→ C(0,1)(G2)C0(A1),

C(0,0,1)(C3) 7→ C(0,1)(G2)C1(A1);therefore

X1 7→ Y1,

X2 7→ Y2,

X3 7→ Y2Y3.

Example of reduction: Let C(a,b,c)(Y ) := C(a,b)(Y1, Y2) be the polynomials of the subalgebra

C2, and Chebyshev polynomials Tm(Y3), m = 1, 2, . . . correspond to the Lie group A1.

C(1,0,1)(C3) = X1X3 − 2X2 7→ Y1Y2Y3 − 2Y2 = C(1,1)(Y )Y3 + 2Y1Y3 − 2Y2;

C(2,0,0)(C3) = X21 − 2X2 − 6 7→ Y 2

1 − 2Y2 − 6 = C(2,0)(Y )− 2.

Reduction from C3 to A2.

Pr =

(1 1 20 1 0

);

C(1,0,0)(C3) 7→ C(1,0)(A2),

C(0,1,0)(C3) 7→ C(1,1)(A2),

C(0,0,1)(C3) 7→ C(2,0)(A2);

thereforeX1 7→ Y1,

X2 7→ Y1Y2 − 3,X3 7→ Y 2

1 − 2Y2.

Example of reduction: In this case, C(a,b,c)(Y ) := C(a,b)(Y1, Y2) are polynomials of the subalge-bra A2.

C(1,0,1)(C3) = X1X3 − 2X2 7→ Y 31 − 4Y1Y2 + 6 = C(3,0)(Y )− C(1,1)(Y );

C(2,0,0)(C3) = X21 − 2X2 − 6 7→ Y 2

1 − 2Y1Y2 + 6 = C(2,0)(Y )− 2C(1,1)(Y ) + 2Y2.

Reduction from C3 to A1.

Pr =(5 8 9

);

C(1,0,0)(C3) 7→ C5(A1),

C(0,1,0)(C3) 7→ C8(A1),

C(0,0,1)(C3) 7→ C9(A1);therefore

X1 7→ Y 5 − 5Y 3 + 5Y,X2 7→ Y 8 − 8Y 6 + 20Y 4 − 16Y 2 + 2,X3 7→ Y 9 − 9Y 7 + 27Y 5 − 30Y 3 + 9Y.

Example of reduction:

C(1,1,0)(C3) = X1X2−3X3−4X1 7→Y 13−13Y 11+62Y 9−129Y 7+97Y 5+20Y 3−37Y = T13(Y )−3T9(Y )−4T5(Y )+T3(Y );

C(1,0,1)(C3) = X1X3−2X2 7→Y 14−14Y 12+77Y 10−212Y 8+310Y 6−235Y 4+77Y 2−4 = T14(Y )−2T8(Y )+T4(Y ).

21

7 Concluding remarks

Given the branching rules for polynomials, their applications now can be contemplated, andconceivably uncommon applications can be stimulated. When changing orbit functions to poly-nomials, the substitution wipes out explicit dependence on group parameters. In spite of that,the structure of polynomials of a given family ‘knows’ about its Lie group. It is uniquely deter-mined by that group.

Perhaps the most important exploitation of Chebyshev polynomials of one variable is theoptimal interpolation of functions, and evaluation of their integrals. We know [10] that theseresults extend to polynomials of groups of type An. Clearly the other types of simple Lie groupsneed to be investigated as well.

The relation between Chebyshev and Jacobi polynomials is known in one variable. In multi-variate context this relation can be established using the fact that our polynomials in n variablesare modification of the monomial symmetric polynomials and the last ones are the building blocksof Jacobi polynomials in n variables (see Sec. 11 of [6]).

Acknowledgements

JP gratefully acknowledges support of this research by the Natural Sciences and EngineeringResearch Council of Canada and by the MIND Research Institute of Santa Ana, California. Allauthors express their gratitude for the hospitality extended to them at the Doppler Institute ofthe Czech Technical University, where most of the work was done.

References

[1] Bremner M.R., Moody R.V., Patera J., Tables of dominant weight multiplicities for repre-sentations of simple Lie algebras, Marcel Dekker, New York 1985, 340 pages.

[2] Dunkl Ch., Xu Yu., Orthogonal polynomials of several variables, Cambridge UniversityPress, New York, 2008, 408 pages.

[3] Hrivnak J., Patera J., On discretization of tori of compact simple Lie Groups, J. Phys. A:Math. Theor., 42 (2009), 385208, 26 pages, math-ph/0905.2395.

[4] Iliev P., Xu Yu., Discrete orthogonal polynomials and difference equations of several vari-ables, Adv. in Appl. Math. 212, (2007), 1–36.

[5] Kass S., Moody R.V., Patera J., Slansky R., Affine Lie algebras, weight multiplicities,and branching rules, Vol.1 and Vol.2, (Los Alamos Series in Basic and Applied Sciences,University of California Press, Berkeley, 1990).

[6] Klimyk A., Patera J., Orbit functions, SIGMA 2 (2006), 006, 60 pages, math-ph/0601037.

[7] Klimyk A., Patera J., Antisymmetric orbit functions, SIGMA 3 (2007), 023, 83 pages,math-ph/0702040v1.

[8] Koornwinder T.H. Orthogonal polynomials in two variables which are eigenfunctions of twoalgebraically independent partial differential operators I-IV, Nedrl. Akad. Wetensch. Proc.Ser. A., 77, 36 (1974) 48-66, 357-381.

[9] Larouche M., Nesterenko M., Patera J., Branching rules for orbits of the Weyl group of theLie algebra An, J. Phys, A: Math. Theor., 42 (2009), 485203, 14 pages; arXiv:0909.2337.

[10] Li H., Xu Yu., Discrete Fourier analysis on fundamental domain of An lattice and on simplexin d-variables, arXiv:0809.1079 (2008), 39 pages.

[11] Lidl R., Wells Ch., Chebyshev polynomials in several variables. Journal fur die reine undangewandte Mathematik 255 (1972), 104–111.

22

[12] Macdonald I.G., Some conjectures for root systems SIAM Journal on Mathematical Analysis13 (6) (1982), 988—1007.

[13] Macdonald I.G., Orthogonal polynomials associated with root systems, SeminaireLotharingien de Combinatoire, Article B45a, Strasbourg, 2000.

[14] McKay W.G., Patera J., Tables of dimensions, indices, and branching rules for representa-tions of simple Lie algebras, Marcel Dekker, New York, 1981, 317 pages.

[15] Moody R.V., Patera J., Computation of character decompositions of class functions oncompact semisimple Lie groups, Mathematics of Computation 48, (1987) 799–827.

[16] Moody R.V., Patera J., Orthogonality within the Families of C–, S– and E– Functions ofAny Compact Semisimple Lie Group, SIGMA 2 (2005), 076, 14 pages, math-ph/0512029.

[17] Moody R.V., Pianzola A., Lambda-mappings between the representation rings of Lie alge-bras, Can. J. Math., 35 (1983), 898–960.

[18] Nesterenko M., Patera J., Three dimensional C-, S- and E-transforms, J.Phys.A: Math.Theor., 41 (2008), 475205, 31 pages; arXiv:0805.3731v1.

[19] Nesterenko M., Patera J., Tereszkiewicz A., Orbit functions of SU(n) and Chebyshev poly-nomials, arXiv:0905.2925v2 (2009), 15 pages.

[20] Nesterenko M., Patera J., Tereszkiewicz A. Orthogonal Polynomials of compact simple Liegroups, arXiv:1001.3683v2 (2010), 31 pages.

[21] Patera J., Sharp R.T., Slansky R., On a new relation between semisimple Lie algebras,J. Math. Phys., 21, (1980), 2335–2341.

[22] Patera J., Compact simple Lie groups and theirs C-, S-, and E-transforms, SIGMA 1

(2005), 025, 6 pages, math-ph/0512029.

[23] Rivlin T.J., The Chebyshef polynomials, Wiley, New York, 1974.

[24] Xu Yu., On discrete orthogonal polynomials of several variables, Adv. in Appl. Math. 33,(2004), 615–632.

23

Appendix

There are three simple Lie groups of rank three, namely A3, B3 and C3. In this section wepresent all the numerical information necessary to deal with orbit functions of three variablesand polynomials in three variables. A part of these data were taken from [18], where some moreinformation about groups of ranks two and three can be found.

Cartan matrices:

A2 : C =(

2 −1−1 2

), C2 : C =

(2 −1−2 2

),

A3 : C =(

2 −1 0−1 2 −10 −1 2

), B3 : C =

(2 −1 0−1 2 −20 −1 2

), C3 : C =

(2 −1 0−1 2 −10 −2 2

).

Weyl group orders:

|W (An)| = (n+ 1)!, |W (Bn)| = |W (Cn)| = 2nn!.

Number of points in the Weyl group orbit:

λ A3 B3 and C3

(⋆, ⋆, ⋆) 24 48λ A2 C2 G2 (⋆, ⋆, 0) 12 24

(⋆, ⋆) 6 8 12 (⋆, 0, ⋆) 12 24(⋆, 0) 3 4 6 (0, ⋆, ⋆) 12 24(0, ⋆) 3 4 6 (⋆, 0, 0) 4 6

(0, ⋆, 0) 6 12(0, 0, ⋆) 4 8

Highest roots:

ξA3 = α1 + α2 + α3, ξB3 = α1 + 2α2 + 2α3, ξC3 = 2α1 + 2α2 + α3.

Fundamental regions:

F (A3) = {0, ω1, ω2, ω3}, F (B3) = {0, ω1,12 ω2,

12 ω3}, F (C3) = {0, 12 ω1,

12 ω2, ω3}.

Discretizations of fundamental regions:

FM (A3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ Z

≥0, s1 + s2 + s3 ≤ M},

FM (B3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ Z

≥0, 2s1 + 2s2 + s3 ≤ M},

FM (C3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ Z

≥0, s1 + 2s2 + 2s3 ≤ M}.

FM (A3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ N, s1 + s2 + s3 ≤ M

},

24

FM (B3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ N, 2s1 + 2s2 + s3 ≤ M

},

FM (C3) ={ s1

Mω1 +

s2

Mω2 +

s3

Mω3 | M,s1, s2, s3 ∈ N, s1 + 2s2 + 2s3 ≤ M

}.

Number of points in FM and FM :

|FM (An)| =(M + n

n

);

|F2k(Bn)| = |F2k(Cn)| =(n+ k

n

)+

(n+ k − 1

n

), k ∈ N;

|F2k+1(Bn)| = |F2k+1(Cn)| = 2

(n+ k

n

), k ∈ N.

|FM (G)| =

0 for M < h,

1 for M = h,

|FM−m| for M > h,

where h is the Coxeter number.

Weight functions f and f for discrete orthogonality of C- and S-polynomials:

f(A2)=|Wu(A2)|J(u), where J(u)−1=4π2√

u21u22−4u31−4u32+18u1u2−27.

f(C2)=|Wu(C2)|J(u), where J(u)−1=4π2√

u21u22−4(u41+4u21−u32−8u22−16u2+6u21u2).

f(A3)=|Wu(A3)|

8π3√

|S(u)|and f(A3)=

4!√

|S(u)|8π3 , where

S(u) = 256−27u41+144u21u2−128u22−4u21u32+16u42−192u1u3+18u31u2u3−80u1u

22u3−6u21u

23

+144u2u23+u21u

22u

23−4u32u

23−4u31u

33+18u1u2u

33−27u43.

f(B3)=|Wu(B3)|

8π3√

|S(u)|and f(B3)=

233!√

|S(u)|8π3 , where

S(u) = (16+4u2−u23)(1728+1728u1+432u21−32u31−16u41+864u2+576u1u2+72u21u2−8u31u2+108u22+36u1u

22−u21u

22+4u32−432u23−216u1u

23+4u31u

23−108u2u

23−18u1u2u

23+27u43).

f(C3)=|Wu(C3)|

8π3√

|S(u)|and f(C3)=

233!√

|S(u)|8π3 , where

S(u) = (u3−2u2+4u1−8)(8+4u1+2u2+u3)(u21u

22−4u32−4u31u3+18u1u2u3−27u23).

Coxeter numbers:

h =

{n+ 1 for An

2n for Bn and Cn.

Level vectors:

Lie group level vector

A2 (2, 2)

C2 (3, 4)

A3 (3, 4, 3)

B3 (6, 10, 6)

C3 (5, 8, 9)

25

Weyl group orbits:

Below we present diagrams of the Weyl orbits of groups A2, C2, A3, B3 and C3, each diagramcontains only half of all its points, the rest of points can be obtained by means of change of signin front of each coordinate of the diagram points.

(a, b)

r1yysss

sssss

ss

r2%%KKKKKKKKK

(−a, a+b) (a+b,−b)

(a, b)

r1xxqqqqqqqqqqq

r2&&MMMMMMMMMMM

(−a, a+b)

r2

��

(a+2b,−b)

r1

��

(a+2b,−a−b) (−a−2b, a+b)

W(a,b)(A2) W(a,b)(C2)

Figure 6: Weyl group orbit of a generic point λ = (a, b, c)ω in the case of group A3.

26

Figure 7: Weyl group orbits: a) W(a,b,c)(B3) and b) W(a,b,c)(C3).

27